On a Special Class of Hyper-Permutahedra

Minkowski sums of simplices in Rn form an interesting class of polytopes that seem to emerge in various situations. In this paper we discuss the Minkowski sum of the simplices ∆k−1 in Rn where k and n are fixed, their flags and some of their face lattice structure. In particular, we derive a closed formula for their exponential generating flag function. These polytopes are simple, include both the simplex ∆n−1 and the permutahedron Πn−1, and form a Minkowski basis for more general permutahedra.